 Phys. Lett. B412 (1997) 350



Towards systematic near-threshold calculations
in perturbative QFT

Fyodor V. Tkachov

Institute for Nuclear Research of Russian Academy of Sciences
60th October Ave. 7a, Moscow 117312 Russia



For any near-threshold asymptotic regime and for any Feynman diagram (involving loop and/or
phase space integrals), a systematic prescription for explicitly constructing all-logs, all-powers
(all-twists) expansions in perfectly factorized form with explicit integrals for coefficients, is pre-
sented. The distribution-theoretic nature of the method of asymptotic operation employed allows
treatment of totally exclusive phase space in the same manner as loop integrals.



Introduction 1 ticity properties of Feynman diagrams; cf. the discussion
in [12].
The purpose of this Letter is to present a summary of sys- The central difficulty of constructing asymptotic expansions
tematic recipe for construction of asymptotic expansions of of Feynman diagrams in masses and momenta is that formal
Feynman diagrams near threshold values of kinematic vari- Taylor expansions of integrands possess non-integrable singu-
ables. The recipe lays a foundation, heretofore lacking, for sys- larities localized on variously intersecting manifolds in the
tematic higher-order calculations as well as for all-order con- space of integration momenta. The key observation (from
struction of asymptotic expansions in operator/effective La- which the entire theory of AO unfolds in a logical manner) is
grangian form. The range of applicability of the recipe includes that the difficulty is a manifestation of the distribution-
many concrete applications such as near-threshold production theoretic nature of the expansion problem [1]4, and that the
of, say, electron-positron pair in QED (and many similar prob- crucial mathematical task is to find expansion of the integrand
lems in the Standard Model); the small-x problem in deeply in the sense of distributions5. Asymptotic operation is a pre-
inelastic scattering; etc. scription that yields such an expansion for a given integrand (=
The recipe is a culmination of the development of the tech- product of propagators).
nique of asymptotic operation (As-operation or simply AO) The structure of AO is fixed by the extension principle
which has been the driving force behind the continuous prog- [1; 15] -- a very general but essentially simple proposition re-
ress in the systematic1 studies of asymptotic expansions of lated to the well-known Hahn-Banach theorem (see e.g. [17]).
Feynman diagrams in masses and momenta since before 1982 The resulting prescription is, roughly, as follows (see also
[1]2. The Euclidean variant of AO yielded powerful calcula- Sec. 2 below):
tional formulas for the short-distance expansion in the MS
scheme [3; 4] and for mass expansions [5-8] that were used in (i) The formal expansion of the integrand should be supple-
a number of NNLO calculations in QCD and the Standard mented with counterterms -- linear combinations of
Model (e.g. [9-11]). The non-Euclidean extension of AO pre- (derivatives of) -functions with coefficients that depend on
sented below is intended to play the same role with respect to the expansion parameter (cf. below Eq. 2.9).

 v3 7 Nov 1998 the near-threshold problems as the Euclidean variant did with (ii) Concrete integral expressions for the coefficients valid
respect to those calculations. within the precision of expansion are obtained from the so-
arXiv: v3 7 Nov 1998 Roughly speaking, Euclidean regimes correspond to cases called consistency conditions [18; 15].
when some masses and momenta are componentwise larger (iii) A fine-tuning of the consistency conditions to achieve
than others.3 On the other hand, one deals with a truly non- purely power-and-log dependence of the coefficients of coun-
Euclidean regime when some momenta have both large and terterms on the expansion parameter (the property of perfect
small components, and the large components approach special factorization [3]).6
values that result in a non-trivial overlap of singularities of dif- After the counterterms of AO are found, obtaining the corre-
ferent factors in the momentum-space integrand. Such situa- sponding all-order operator expansions is a matter of more or
tions correspond to thresholds from the point of view of analy- less straightforward combinatorics (cf. the Euclidean case [8]).


4 This observation was influenced by Bogoliubov's analysis of the UV
1 "Systematic" here means all-logs, all-powers treatments complete with problem [13; 14].
explicit calculational formulas in a form maximally suitable for practical 5 For precise definitions see [15; 16].
applications. 6 Note that the deterministic logic of AO leads one step-by-step towards the
2 For a review and complete references to the original publications, as well solution. Contrast this with the BPHZ-type methods where one, in fact,
as for comparison with the conventional methods such as the BPHZ theory needs to guess the result (the forest formula) as a whole in order to proceed
and the technique of leading logarithmic approximation, see [2]. to, say, formal proofs. For instance, the BPHZ-type interpretation of the
3 To this class also belong degenerate cases with external large momenta general Euclidean expansions needed to borrow the explicit results that
are fixed on mass shell which simply corresponds to complex-valued large had been previously obtained within the framework of AO (for details
external momenta from the point of view of Euclidean space. see [19]).


F.V.Tkachov Near-threshold expansions Phys. Lett. B412 (1997) 350  2





Thus, the analytical focus of the entire theory of asymptotic called homogenization -- a secondary expansion that compl e-
expansions of Feynman diagrams is in finding the coefficients ments the consistency conditions for the coefficients of AO.
of the counterterms of AO in a form best suited for applica- The homogenization splits those coefficients into pieces with
tions. The point is that, in general, asymptotic expansions are strict power-and-log dependence on the expansion parameter
not unique (which is reflected in an arbitrariness of the expres- (powers needing not be integer). Below we expand the sce-
sions for coefficients of AO obtained from consistency condi- nario of [2] by presenting explicit rules for the homogeniza-
tions). Uniqueness is restored [15], however, if one imposes on tion.8
expansions the requirement of perfect factorization [3] -- It is remarkable that a self-contained analytical recipe for an
which at the level of individual diagrams stipulates that the individual diagram can be summarized in a rather compact
expansions run in pure powers and logarithms of the expansion universal form for arbitrary non-Euclidean regimes despite
parameter. Apart of its importance for applications their larger analytical variety than in the Euclidean case. Such
(determination of power-suppressed corrections is impossible a description would not be possible without using the language
without it [20]), the uniqueness of power-and-log expansions of AO. However, there is both an increasing familiarity among
has a number of technical advantages: physicists with the technique of AO, and a tendency to use it in
(i) AO commutes with multiplications by polynomials thus various non-standard physical problems due to its power and
allowing one to ignore complications due to non-scalar flexibility (cf. [22-26])9. On the other hand, not all such works
particles; have been equally successful.10 Therefore, it would be useful
-- prior to a more complete treatment which would require a
(ii) expansions inherit all algebraic properties of the initial substantially longer text -- to give a summary of the procedure
integrands (such as the Ward-Takahashi-Slavnov-Taylor iden- in a form suitable for calculations that can also serve as a
tities); starting point for derivation of all-order operator-form expan-
(iii) maximal calculational simplifications; (iv) considerable sions for various regimes.
advantages for a regularization-independent treatment [16; 21], Before we proceed to formulas, a few remarks are in order.
which is potentially important for supersymmetric models. (i) One sometimes uses the term Minkowski space regimes
The consistency conditions have the form of integrals in- (cf. [27]). However, Minkowski space per se allows both
volving some of the propagators of the original Feynman inte- Euclidean regimes (cf. their treatment in Minkowski space in
grand as well as a test function whose behavior at the singular [12]), and a more complex class of non-Euclidean, or near-
point is fixed but which is otherwise arbitrary (cf. Eq. 2.12). threshold regimes. I emphasize the distinction in order to avoid
For the Euclidean case, it was found [18; 15] that it is suffi- confusion due to vagueness of terminology and argumentation
cient to replace such test functions with suitable polynomials in many publications on the subject.
(to be understood as an appropriate limiting procedure; for a (ii) In some recent publications the term "threshold expan-
rigorous treatment see [16; 21]). It turned out that such a re- sion" was misused to denote cases with kinematic parameters
placement modifies the expressions only within the precision set exactly at threshold values with expansions running with
of the expansion (which is always allowed), whereas the sim- respect to some other parameter (e.g. internal mass), or degen-
ple scaling of the integrand in momenta and masses ensures erate thresholds tractable by Euclidean methods. In this Letter
the power-and-log dependence of the resulting expression on we consider true non-degenerate thresholds (including non-
the expansion parameter. This allowed us to include into sys- zero ones) intractable by Euclidean methods, and expansions in
tematic consideration the entire class of Euclidean asymptotic a parameter that measures closeness to such a threshold.
regimes [18] and subsequently to expand the scope of operator- (iii) The method of AO considers integrands in momentum
expansion methods to such regimes. space as distributions prior to integration. It is therefore ideally
suited for studying physical problems where integration over
As was emphasized from the very beginning [18; 15; 2], the the phase space of final state particles should not be performed
scenario of AO is completely general and by no means limited in an explicit fashion. Indeed, the -functions that describe the
to Euclidean cases. The specifics of the non-Euclidean regimes 2 2
is that the singularities of integrands are localized on non- phase space -- e.g. ( p ) ( p - m )
0 etc. -- are, from the
linear manifolds (light cones and mass shells) and that non- distribution-theoretic point of view, equally acceptable factors
zero finite limiting values for external momenta break the alongside the standard causal propagators ( p2 - m2 + i )-1
0 .
usual scale invariance of integrands. As a consequence, the This opens a prospect for a systematic treatment of e.g. the
simple trick that yielded power-and-log dependence in the problem of power corrections in jet-related shape observables
Euclidean case is no longer sufficient. However, it is important in the context of precision measurements of
to understand that the consistency conditions are obtained from S according to
first principles without any restricting assumptions, and there- the theoretical scenario outlined in [28] in analogy with the
+ -
fore possess all the flexibility to accommodate any additional case of total cross section of e e hadrons where the op-
requirement that one may lawfully impose -- in particular, to
perform an appropriate fine-tuning to achieve the required 8 Ref.[2] outlined the worst-case scenario for non-Euclidean AO because a
power-and-log dependence on the expansion parameter. universal description of its structure was not yet available. Thus, the com-
binatorial complexity due to the homogenization seems now to be less se-
The resulting problem and its solution were identified in vere -- at least for some asymptotic regimes -- than anticipated then.
[2]: the problem consists in occurrence of the so-called oscu- 9 The integral version of AO [6; 8] (cf. also its first special case -- the
lating singularities7, whereas the solution is given by the so- formulas for OPE coefficient functions in the MS scheme [3; 4]) is useful
in situations where efficient automation is necessary, such as higher-order
7 i.e. singularities whose singular manifolds touch rather than intersect in a calculations etc.
general fashion; for a detailed discussion see [12]. When a mixture of os- 10 For instance, the description of AO in ref.[25] is incorrect whereas the
culating and transverse intersections occurs, simple uniform scaling rules authors of [26] independently found via AO a correct treatment of a con-
for description of the strength of the singularity no longer suffice. crete near-threshold problem.


F.V.Tkachov Near-threshold expansions Phys. Lett. B412 (1997) 350  3





erator-product expansion can be used for that purpose [20]. UV renormalization is assumed to be performed in
Problems with exclusive phase space occur on a massive scale a massless scheme of the MS type and is treated following
in physical applications.11 The various cuts used for event se- [33; 34; 21] as a subtraction from momentum space integrand
lection are equivalent to various weights in phase space inte- of its asymptotic terms (in the sense of distributions) for
grals, which means that the corresponding matrix elements p (see [15; 16] for an exact interpretation of the large-p
squared are effectively treated as distributions in the momenta limit involved). For practical purposes, it is sufficient to
of the final state particles -- a perfect setting for application of employ the MS scheme [35] (or any of the massless renormali-
the distribution-theoretic technique of AO. zation schemes), and treat unrenormalized UV-divergent
integrals formally as convergent in the usual fashion (cf. the
Description of the method 2 prescriptions of the Euclidean AO [15; 8]; a rigorous treatment
of why this is possible is given in [16; 21]).
The prescriptions of AO are best described as a formal deri-
vation rather than a final formula or theorem. The reason is Expansion parameter and asymptotic regimes 2.3
that every step of the derivation has a simple concrete meaning
enabling one to control correctness of formulas in each con- The expression 2.2 depends on external parameters such as
crete situation, whereas blindly using a cumbersome final for- masses, momenta of incoming particles etc. It is assumed that
mula may result in gross errors.12 some of the momentum components and/or masses are small
We follow the notations of [15; 16] and focus here only on compared to others. The small parameter (one with respect to
the most difficult -- analytical -- aspect of the expansion which the expansion is to be performed) will be denoted as .
problem; the diagrammatic interpretation depends on a con- In general, one assumes that some of the external parameters
crete asymptotic regime and is a much simpler (combinatorial) -- masses or momenta -- tend to specific values (zero or not),
issue anyway. We describe AO in the form with an intermedi- and that the differences between the external parameters and
ate regularization similarly to how the Euclidean case was their limiting values are of order (extension to cases with
treated in [15]. It is not difficult to convert the formulas into a several scales of the form n is straightforward). The limiting
regularization-independent form similar to [16] (further details values of external momenta need not be zero componentwise.
specific to the non-Euclidean are given in [30]). In general, we Some examples of the constructs mentioned in the description
have in view a combination of dimensional [31] and analytical given below, are presented in Sec. 3 (for further examples see
[32] regularizations; the latter may be needed in the cases [23] and [36]).
when the dimensional regularization alone is insufficient (cf. Our prescriptions are valid irrespective of what kind of
the example of [23]). threshold the chosen asymptotic regime corresponds to -- pe r-
haps, none at all in which case there will simply be no singu-
The expression to be expanded 2.1 larities requiring addition of non-trivial counterterms.
The collection of all integration momenta (loop and phase Note also the following rule for the problems with explicit
space) is denoted as p . The integrand of the diagram to be ex- phase space: If some phase space momentum components are
panded in a small parameter is represented as follows: to be treated as small, i.e. O( ) , they should be made O( )
1 by

G( p) = g( p) , g( p)
= el (p, )j , 2.2 appropriate rescaling before applying the procedures of AO.
g G g g

Formal expansion 2.4
where =  -
g (z) (z i )
0 1 or (z) (each such factor will be re-
ferred to as "propagator"); l p The construction of AO begins with the formal (usually, but
g ( , ) is a second order polyno- not necessarily, Taylor) expansion of the integrand in powers
mial of the momenta p and masses (first order polynomials are of : G( p) T G( p)
o . Each factor is expanded separately,
allowed as a special case; cf. gauge boson propagators in non- the results are formally multiplied and reordered in increasing
covariant gauges). Note that we allow as factors causal propa- powers of . The terms of the resulting series possess, in gen-
gators, their complex conjugates, and phase-space -functions eral, non-integrable singularities which have to be examined in
(for simplicity, the -functions of phase space factors are the geometrical and analytical aspects.
omitted; to include them, it is sufficient to modify g accord-
Geometric classification of singularities
ingly in all formulas; cf. the example in Sec. 3.1). Various of the formal expansion 2.5
polynomials that may occur in the numerator (due to non-scalar
particles and interactions with derivatives) are ignored because Each formal expansion T o g( p) of each factor g( p) of
AO commutes with multiplication by polynomials [15]. The the initial product 2.2 is singular on the manifold
expression 2.2 comprises as special cases loop and unitarity g de-
diagrams. scribed by l p =
g ( , )
0 0 . The aggregate singular manifold of

T o G( p) is
gG g . In general, the latter is singular in the

sense of differential geometry, so one splits it into non-singular
11 I am indebted to S. Jadach for explaining to me this point. components (labeled by an index ; each such component
12 Incidentally, Collins et al. [25] attempted to describe a non-Euclidean is a smooth open manifold). To each component there corre-
example of AO. The result is a bizarre text which makes very little sense
beyond vaguely echoing the discussions I had with John Collins during my sponds a subproduct F p G p
( ) ( ) (but unlike the Euclidean
three visits to Penn State (I believed we were discussing application of the case, here different
technique of AO to the Sudakov problem; cf. [23]). Several of the formu- may correspond to the same sub-
las of [25] (together with the accompanying textual descriptions) are sim- product; e.g. the apex of the light cone and its two cones corre-
ply incorrect. This calls for a critical reexamination of the "proofs" of spond to the same propagator). F ( p) ( p) contains all the
QCD factorization theorems, which I intend to do elsewhere [29].


F.V.Tkachov Near-threshold expansions Phys. Lett. B412 (1997) 350  4





factors from G that are singular everywhere on . Denote as and all l p
g ( , )
0 with L p
g ( ) 0 are independent of Z . Then X

G \ ( p) the product of all factors from G that do not belong and Y scale with 2 , and Z scales with .14 As a simple check,
to . One may say that represents a "subgraph"; its corr e- this rule correctly yields a uniform scaling in all components of
sponding product of propagators F ( p) ( p
) is determined p and both when all l p
g ( , ) are linear functions, and when

uniquely.13 they are all purely quadratic functions (the Euclidean case).

Analytical structure of singularities 2.6 Power counting15 One performs the power counting to de-
termine the strength of singularity in each term of T o G( p)
Here we have to set rules for power counting and -- simu l- near generic points of each
taneously -- to define the so-called homogenization -- a se c- . For that, one drops from de-
ondary expansion needed in non-Euclidean situations to reduce nominators (and/or arguments of -functions) all but those
the coefficients of counterterms of AO to power-and-log form. components of l p
g ( , )
0 that scale with the lowest power of ,
The discussion below is in the context of a given subgraph . and introduces into the numerator the factor
One considers a general point p0 of and the behavior of 2dim X +2dimY+dimZ that corresponds to the scaling of the in-
F p
( ) when p p0 along directions that are transverse to tegration measure. Collecting all powers of one obtains an
-
. The components of p that are tangential to are overall factor where can be appropriately called the
"spectators" and are to be ignored. After introducing appropr i- singularity index of the subgraph (e.g. = 2 corresponds to
ate local coordinates near p0 and redefinitions, we may as- quadratic divergence etc.).
sume in what follows that p does not have spectator compo-
nents and that = { }
0 . Homogenization 2.7

The homogenization parameter
Scaling Scale p ni
p
n is introduced as fol-
i i , so that:
lows: (i) scale l p g
as described; (ii) drop the over-
(i) all scaling exponents n are positive integer; g ( , ),

(ii) for each g , l p all ng ; (iii) replace with . The operation of homogeni-
g ( , ) scales as
n zation (denoted as H
g ) is as follows: (i) introduce as just
lmain ( p
, ) + O(
g ) , and the scaling exponent for is the
described; (ii) expand in ; (iii) set 1 in the result.
minimal value ensuring this for given ni ; H is meant to be applied to integrals similar to those we set
(iii) if main main =
g is the manifold on which l ( p
g , )
0 0 , then out to expand from the very beginning (see below Eq. 2.12).
main
-- defined as Therefore, the expansion in H
g main
g i.e. the set of all p such that must in general be performed
in the sense of distributions -- requiring the entire machinery
l main( p =
g , )
0 0 for all g -- coincides with = { }
0 . The of AO with another level of homogenization etc. Since at each
latter means, in particular, that the collection of all step the expansion problem simplifies (dimensionality of the
l main( p integration space is reduced), the recursion stops correctly after
g , ) , g depends on all components of p .
a finite number of levels of homogenization. In many interest-
These properties ensure that each step of our expansion ing cases, however, non-trivial singularities requiring coun-
procedure is a mathematically correct transformation. A re- terterms do not occur and this expansion degenerates into a
markable fact is that the scaling satisfying (i)(iii) need not be simple Taylor expansion.
unique (e.g. in the case of radiative corrections to the example The occurrence of secondary expansions with their corres-
of Sec. 3.1). The beautiful mathematical mechanism of how dif- ponding homogenizations is behind the mechanism that en-
ferent scalings result in the same final answer, is mentioned sures independence of final results of the choice of scaling
below after the definition of homogenization -- we have a l- (provided the latter satisfies the three conditions mentioned in
ready had opportunities [15] to emphasize a remarkable stabil- Sec. 2.6): non-trivial counterterms for secondary homogeniza-
ity of the method of AO that yields correct results even in tions yield expressions that correspond to alternative scalings,
counterintuitive cases as long as one applies it in a systematic whereas the "formal" part of the homogenization expansion
manner. Different correct scalings do differ in the number of yields zero by explicit integration. Demonstrating this mecha-
intermediate steps leading to the (same) final result. An opti- nism in detail requires a much more detailed exposition than
mal definition is as follows: Split l ( p, )
0 L ( p) + L( p
g g g ) we can afford here.
with L p = O p 2
g ( ) ( ) , L p = O p
g ( ) ( ) . Then split

p ( X ,Y, Z) so that all L p
g ( ) depend on, and only on X


13
The pinch/non-pinch classification of singularities (the usual starting 14
point of the conventional analyses; cf. e.g. [27]) is actually irrelevant for For general non-quadratic (cubic etc.) functions one may recur to a
the analytical study of the singularities in general, and for the construction technique based on Newton's polyhedra similar to that used in the theory
of AO in particular: the non-pinched singularities simply correspond to of singularities of differential mappings [37].
parts of singular manifolds where the corresponding counterterms nullify 15 This is regarded as a technical problem in the context of the theory of
(which can be deduced e.g. directly from their expressions). Such a nullifi- QCD factorization theorems [27], esp. in the case of non-leading power
cation is rather accidental from the point of view of analytical structure of corrections and mixed soft/collinear singularities. Our rules should settle
singularities. the issue (see also [30]).


F.V.Tkachov Near-threshold expansions Phys. Lett. B412 (1997) 350  5





Structure of asymptotic operation 2.8 last expression in 2.12 is determined by scaling out using the
scaling rules for already explained.
Recall the general formula for AO [15]: An important point to remember is that the construction of
As o G( p) = T o G( p) + E ( p)
 T
o G \ ( p) . 2.9 asymptotic expansions (including AO) is always, strictly
speaking, carried out for a particular finite precision O N
( ) .
It is rather natural: each singularity of the formal expansion This implies that one subtracts only the series to that precision
T o G( p) receives a counterterm E ( p) localized on the cor- in the integrand of 2.12. Correspondingly, the counterterm is
responding singular manifold . So, the summation runs also defined at that step only within that precision, whereas
expansion implied by H -- if carried too far -- would gene r-
over all subgraphs , and E ( p) have the form
ate terms of excessive precision O N n
( + ) -- terms that would
E ( )
p = E

( ) 
( p)
, , , 2.10 also be divergent in the UV region! One can verify by power
counting, however, that all the contributions of precision
where summation runs over a complete set of -functions lo- O N
( ) are exactly those whose (formal) UV convergence is
calized on , and E
, ( ) in general are -- unlike the ensured by the subtraction. In the end one can forget about the
Euclidean case where each E
, ( ) is proportional to one in- restriction O N
( ) because the formula (the last expression in
teger power of -- series in (non-integer) powers of with 2.12) is independent of it.18
coefficients that are polynomials of log . As was already em- Another important point concerns the diagrammatic inter-
phasized, finding those coefficients is the central analytical pretation of the subtraction in the first expression in 2.12. In
task of the theory of asymptotic expansions of Feynman dia- the Euclidean case, it was shown [34; 21] that the subtractions
grams. of this sort exactly correspond to the standard Bogoliubov UV
R-operation -- in agreement with the fact that the integrals
Consistency conditions for the coefficients 2.11 that occurred there for E, were exactly diagrams with local
The finding of E
, ( ) is performed according to the same operator insertions. In the general non-Euclidean situation,
general recipe as in [15], i.e. via consistency conditions. Sup- there is no universal (i.e. valid for all regimes) operator char-
pose one wishes to construct the coefficients for E acterization for the integrals that occur after the homogeniza-
, ( )
for tion (the expansion H distorts standard propagators in differ-
one subgraph . One assumes that for all < the construc-
ent ways for different asymptotic regimes). But the prescription
tion has been performed (the usual ordering of subgraphs with for their UV renormalization is always determined uniquely by
respect to increasing codimensionality of singular manifolds is the structure of subtractions in 2.12.
assumed here; cf. [15]). An appropriate choice of coordinates
ensures that the singularity one is after is localized at p = 0 . As a last remark, an interesting technical point may be
mentioned. Namely, despite the presence of the additional ex-
Then the coefficients are given by the following formulas (cf.
sec. 12.3 in [15] and eq. (20.5) in [16]): pansion H in Eq.2.12 compared to the Euclidean case, when
one integrates out the -functions similarly to the procedure of
E ( ) = lim X
d Y
d Z
d ( X-2 ,Y-2 ,Z -
z 1
)
, sec. 5.3 of [8] to obtain the AO in integral form, this additional
 P series blends into a single Taylor expansion with the expansion
( p) H o 1 - A
s o
resulting from derivatives of -functions (cf. eq.(5.8) in [8]).
= p
d P ( p) H o ( p, ).
z 2.12 This is easily explained if one notices that the non-Euclidean
prescriptions remain valid in the general non-linear/non-
(The polynomials P form a complete dual set for the - uniform case, so that the "spectator" product G \ can be
functions formally absorbed into and treated as a "deformation" of the
, -- exactly as in the Euclidean case. As o
latter, and when obtaining the AO in integral form, only those
differs from As o by absence of the term with = in the counterterms of the AO would survive that are proportional to
corresponding sum 2.9.) The first expression demonstrates how -functions without derivatives. But then the expansion H
the intermediate cutoff is removed. Notice the asymmetry of
the cutoff which corresponds to the asymmetry of the scaling. would be the only expansion remaining in the end! (Note,
The role of H however, that the expansion implied by H
(a new element compared to the Euclidean affects the entire
case) is to split the coefficient into terms with pure power-and- G -- in contrast with the Euclidean case where only certain
log dependence on (which motivates its definition). The sec- subgraphs are thus expanded; cf. the expansions of "heavy
ond expression takes into account the nullification of the sub-
tracted ("shadow") terms in dimensional/analytic regulariz a- ties by the dimensional/analytical regularization, although now the scaling
tion, which is due to their pure-power behavior under the is non-uniform in different components of p .
(asymmetric) scaling.16,17 The power of for each term in the 17 To obtain the corresponding regularization-independent formula one
~
f o
replaces As r As where ~rf is an appropriate generalization of
16 the corresponding Euclidean subtraction operator; see [16] and [30].
Nullification of integrals of the form d D p p
z -2 = 0 is well-known. It
18 However, a great caution must be exercised when establishing corre-
is due to the fact that the dimensional regularization preserves formal spondence between the terms of the last expression in 2.12 and the singu-
scalings. In the new situation, we are dealing with zero integrals of, very larities of the formal expansion T
- o G . The correspondence is rather
roughly, the form d D-2 p dp c Ap + p
z + + h
2 2 = 0 . The underlying tricky which fact caused some confusion in the literature (see a discussion
reason for their nullification is the same -- preservation of scaling prope r- in [19]).


F.V.Tkachov Near-threshold expansions Phys. Lett. B412 (1997) 350  6





knots" in [8].) An amusing implication is that use of the gen- in Taylor-expanding with respect to the quadratic terms in p2
0
eral non-Euclidean results would actually simplify the con- etc.), we find:
struction of Euclidean AO in integral form.
E D
( ) = d p e2mp - 2
p j e2m( - p ) -
z 2 j
0 0 p
Examples 3
3/
1 2- 1 2-
=   (m ) / ,
b g 3.5
The two examples we consider are related to well-known 4m /
3 2-

simple integrals so that checks are possible, yet involve ex- where = 1 (4 - D) . The final result for 3.2 is as follows:
plicit phase space so that a systematic treatment via any other 2
method would be problematic. In both examples, obtaining dD p w( p) e p2 - m2j e(Q - p)2 - m
z 2
+ + j
higher terms of the expansion is tedious but entirely straight-
forward -- an exercise that is left to an interested reader. Su b- = Q2 - m2 w(m~) + OeQ2 - m2
4 4 j. 3.6
4m
stantially more involved examples will be presented elsewhere
[38; 39]. (The integral of the first term of the formal expansion -- the
first term on the r.h.s. of 3.4 -- is zero for smooth w.) This re-
Threshold Q2 ~ m2
4 for the kinematics Q m + m 3.1 sult is checked by noticing that the case w = 1 corresponds to
the imaginary part of a simple self-energy diagram; cf. the ex-
This example corresponds to a near-threshold creation of a plicit result in [40], eq.(24.5).
pair (e.g. + -
e e by a photon). Remember that asymptotic op-
eration commutes with multiplication by polynomials so that The behavior of 2 2 at s + 3.7
non-scalar numerators are ignored in our example.19 The phase Our second example corresponds to the matrix element
space is represented as squared with exclusive phase space for k + k k + k
+ - + -
d D p w( p) e p2 - m2j e(Q - p)2 - m
z 2
+ + j , 3.2 (where k = k 
  p ) via simplest t -channel exchange of a

scalar particle with momentum p at large s = (k + k )2 . The
where (k 2 m2 ) (k ) 2 2 + -
+ - = (k - m )
0 , Q is the momentum
example is motivated by the large-s /small-x problem in QCD
of the "photon" that decays into the pair, and the arbitrary [41], so all particles (partons) are massless and the external
weight w (corresponding to arbitrary cuts experimentalists may partons are slightly off-shell (time-like). The expression to be
use for event selection) means that the phase space is treated expanded is:
as totally exclusive, so that one essentially deals with the prod-
uct of 's and 's interpreted as a distribution. We are going to 2 2 -
s p
d w( p) e(k + p) j e(k - p) j p
z 2
+ + + - , 3.8
extract the first non-trivial (square root) contribution using the
prescriptions set forth above. where e(k + p)2j = (k0 + p0) 2
+ + + e(k+ + p) j etc., and we
To describe the asymptotic regime we choose have replaced for simplicity the exchange particle's squared
Q = b m
2 + ,0g with 0 . Then propagator p-4 by p-2 (imagine e.g. that there occurred a

= (Q2 - m2) m - (Q2 - m2)2 ( )
m 3
4 4 4 4 + K 3.3 cancellation with p2 from a non-scalar numerator). w is an ar-
bitrary weight that describes cuts, observables etc. The as-
It is convenient to perform a shift p p + m
~, m~ = (m,0) . The
ymptotic regime is described by s with k 2 , p2 O
 = 1
( ) .
formal expansion of the product in then is: Introduce two light-like vectors q such that 2q q 1
+ - = and
e(p + ~m)2 - m2j e( ~m +~ 2 2
+ + - p) - m j k = s q + O(s- /
1 2
  ) . Taking out s from each + and de-
= ep2 + m
2 p - 2
p j 2 2
+ e p j /
0 0 + p - m
2 p -
0 0 noting = -
s 1 2 , rewrite 3.8 as

+
2 - p + m ep2 + 2mp - 2
p j 2 2
e p j K z -
( ) 2 e2 + 2
+ 2
+ 2
j
0 + 0 0 + p - m
2 p - +
0 0 3.4 dp w p p q p k p O( p
+ + + )

The singularity at the point p = 0 is seen from the fact that the e- q
2 p + k 2
+ p2
+ O( 2
p
+ - - )j . 3.9
arguments of the two -functions reduce to the form
The first term of the formal expansion is:
( p ) ( 2
p )
0 , with the second ill-defined in the case of -2
c h c h
higher derivatives (the singularity is regulated by the dimen- dp w( p) 2q p -2q p p + O
z + + + - ( ) . 3.10
sional regularization). To do the power counting according to =
{p = zq ; z
the prescriptions of Sec. 2, one identifies X ( p , ), Y Singularities are localized on   } and
0 p
=
with Z empty. The first product of 's is convergent whereas 0 0
{ } . The two collinear singularities  are similar to
the second one is linearly divergent. In any case, a counterterm those considered in [23]. Following the above rules, one finds
of the form E( ) ( p) is necessary. Evaluating E( ) ac- the appropriate scaling (in the Sudakov coordinates):
2
cording to the given recipe (the homogenization consists simply p 2 p
m m , p p
, (for  the component
pm is spectator and does not scale). One finds that in the
leading order the two singularities are logarithmic, so only -
19 Of course, they do play a role in determining which particular terms functions without derivatives have to be added to 3.10:
from the asymptotic expansion contribute to a particular process at a par- + +
dz c (z, ) ( p - zq ) + dz c (z, ) ( p - zq
z z ) . 3.11
ticular precision with respect to the small parameter. But our aim here is - + + - - -
only to demonstrate the essential analytical mechanism.


F.V.Tkachov Near-threshold expansions Phys. Lett. B412 (1997) 350  7





The corresponding coefficients are easily found: For Monte-Carlo type applications one needs to recast the
results into a regularization-independent form. The correspond-
c (z D
, ) = d p (2 pq + k 2)cz - 2 pq hp-
z 2
 +   m ing prescriptions are derivative from the power-counting rules
of Sec. 2.6 and follow the general pattern of [16; 21]. More
1 - -

~ bmzg ek2j
+  , 3.12 details will be given elsewhere [30].


where A ( A )
0 A
+ = > and we have dropped irrelevant coef- Conclusions 4
ficients. However, when one multiplies the collinear coun-
terterms by the corresponding G \ = (m q p) The recipe for the non-Euclidean (near-threshold) asymp-
m + 2 m , one finds
totic operation given above is a very general one, is valid prac-
that the product vanishes because c (z) ~ z-
 (cf. the case tically for any non-Euclidean asymptotic regime, and for any
with a non-zero mass where a similar product was ill-defined individual Feynman diagram, including unitarity diagrams with
[23]). cut propagators as well as diagrams in non-covariant gauges,
The scaling for the singularity localized at = heavy-quark and non-relativistic effective theories, etc. But de-
0 0
{ } is uni-

form in all components p p
, . The counterterm spite its generality, it relies on few analytical principles, which
is important for calculationists who wish to have a complete
then is c p
0 ( ) ( ) with intellectual control over what they are doing (cf. e.g.
- [22; 24; 26]). In many applications (namely, automated higher-
c D
( ) = d p e2q p + k2
j e-2q p +k2j p 2
z
0 + + + + - - order calculations and derivations of all-order near-threshold

1 2
-
~ . expansions in operator/effective Lagrangian form) one would
3.13
need the rules for AO in integral form similar to the Euclidean
Due to all the -functions, the integrals in this example are ones found in [5-8], that are currently in wide use (cf. [9-11]).
performed very easily. Such rules are not difficult to obtain -- but then the universa l-
ity of formulas is lost and each regime has to be treated sepa-
Finally, the expansion is: rately. Studying the operator form of expansions for various
Eq. 3.8 = Eq. 3.10 + c ( ) w
0 (0) . 3.14 non-Euclidean (near-threshold) regimes and performing the
corresponding calculations will be a topic of active research in
Performing integration in 3.10 with w = 1 one finds that the the coming years [26; 38; 39; 43].
poles -1 in 3.14 cancel (as expected). One sees that the
log contribution is associated with the diffractive region Acknowledgements I am indebted to V.I. Borodulin, G. Jikia
p ~ 0 . and N.A. Sveshnikov for an extremely important support.
I thank D. Broadhurst and P. Nason for pointing out the impor-
If one had normal non-cut propagators instead of the two - tance of studying non-zero thresholds in QED and the Standard
functions in 3.8, the vanishing of collinear counterterms would Model, S. Jadach for emphasizing the importance of exclusive
not occur, which would give rise to the well-known log2 treatment of phase space, A.V. Radyushkin for bibliographic
terms (taking imaginary part eliminates one logarithm). Oth- advice, and F. Berends for an encouraging laugh. This work
erwise the integrals are just a bit more involved, and the cor- was supported in part by the Russian Foundation for Basic
rectness of calculations is easily checked by comparison with Research under grant 95-02-05794.
the well-known analytical results (see e.g. [42]).


F.V.Tkachov Near-threshold expansions Phys. Lett. B412 (1997) 350  8





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